integral of (x/2-1/x+3/(x^3)-4/(x^4))

解答

\int (\frac{x}{2} - \frac{1}{x} + \frac{3}{x ^{3}} - \frac{4}{x ^{4}}) d x

\frac{x ^{2}}{4} - ln ∣ x ∣ - \frac{3}{2 x ^{2}} + \frac{4}{3 x ^{3}} + C

求解步骤

\int \frac{x}{2} - \frac{1}{x} + \frac{3}{x ^{3}} - \frac{4}{x ^{4}} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{2} d x - \int \frac{1}{x} d x + \int \frac{3}{x ^{3}} d x - \int \frac{4}{x ^{4}} d x

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{3}{x ^{3}} d x = - \frac{3}{2 x ^{2}}

\int \frac{4}{x ^{4}} d x = - \frac{4}{3 x ^{3}}

= \frac{x ^{2}}{4} - ln ∣ x ∣ - \frac{3}{2 x ^{2}} - (- \frac{4}{3 x ^{3}})

化简

= \frac{x ^{2}}{4} - ln ∣ x ∣ - \frac{3}{2 x ^{2}} + \frac{4}{3 x ^{3}}

解答补常数

= \frac{x ^{2}}{4} - ln ∣ x ∣ - \frac{3}{2 x ^{2}} + \frac{4}{3 x ^{3}} + C

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